Page 41 - Probability, Random Variables and Random Processes

CHAP. I] PROBABILITY

Let Ai be the event that component si functions. Then
P(Ai) = 1 - P(Ai) = 1 - pi
Let A be the event that the system functions. Then, since A,'s are independent, we obtain

1.59. A system consisting of n separate components is said to be a parallel system if it functions when
at least one of the components functions (Fig. 1-17). Assume that the components fail indepen-
dently and that the probability of failure of component i is pi, i = 1, 2, . . . , n. Find the probabil-
ity that the system functions.

Fig. 1-17 Parallel system.

Let Ai be the event that component si functions. Then

Let A be the event that the system functions. Then, since A,'s are independent, we obtain

1.60. Using Eqs. (1.85) and (1.86), redo Prob. 1.34.
From Prob. 1.34, pi = 4, = 1, 2, 3, 4, where pi is the probability of failure of switch si. Let A be the
i
event that there exists a closed path between a and b. Using Eq. (1.86), the probability of failure for the
parallel combination of switches 3 and 4 is
P34 = P3 P4 = (+)(a) == a

Using Eq. (1.85), the probability of failure for the combination of switches 2, 3, and 4 is
p234 = 1 - (1 - 4x1 - i) =; 1 - 3 =
8 8
Again, using Eq. (1.86), we obtain

1.61. A Bernoulli experiment is a random experiment, the outcome of which can be classified in but
one of two mutually exclusive and exhaustive ways, say success or failure. A sequence of Ber-
noulli trials occurs when a. Bernoulli experiment is performed several independent times so that
the probability of success, say p, remains the same from trial to trial. Now an infinite sequence of
Bernoulli trials is performed. Find the probability that (a) at least 1 success occurs in the first n
trials; (b) exactly k successes occur in the first n trials; (c) all trials result in successes.
(a) In order to find the probability of at least 1 success in the first n trials, it is easier to first compute the
probability of the complementary event, that of no successes in the first n trials. Let Ai denote the event